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Topic: The ambiguity of 0^0 on N
Replies: 106   Last Post: Sep 29, 2013 10:06 AM

 Messages: [ Previous | Next ]
 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 19, 2013 10:24 PM

On Thursday, September 19, 2013 7:36:57 PM UTC-4, fom wrote:
> On 9/19/2013 1:41 PM, Dan Christensen wrote:
>

> > On Thursday, September 19, 2013 1:16:44 PM UTC-4, Peter Percival wrote:
>
> >> Dan Christensen wrote:
>
> >>
>
> >>
>
> >>
>
> >>>> There is a simple definition of exponentiation on N, based on
>
> >>
>
> >>>>
>
> >>
>
> >>>> cardinal arithmetic. It doesn't have any special cases, but
>
> >>
>
> >>>>
>
> >>
>
> >>>> 0^0 = 1 falls out of it quite naturally:
>
> >>
>
> >>>>
>
> >>
>
> >>>>
>
> >>
>
> >>>>
>
> >>
>
> >>>> |A|^|B| = |{f | f:B->A}|
>
> >>
>
> >>>>
>
> >>
>
> >>>>
>
> >>
>
> >>>>
>
> >>
>
> >>>> In English, the cardinal number of set A raised to the power
>
> >>
>
> >>>>
>
> >>
>
> >>>> of the cardinal number of set B is the cardinal number of the
>
> >>
>
> >>>>
>
> >>
>
> >>>> set of all functions from B to A. No exceptions, no special
>
> >>
>
> >>>>
>
> >>
>
> >>>> cases.
>
> >>
>
> >>>>
>
> >>
>
> >>>
>
> >>
>
> >>> Thanks, but as with any analogy, it may not be perfect.
>
> >>
>
> >>
>
> >>
>
> >> It's not an analogy, it's a fact. I've just taken off my bookshelves
>
> >>
>
> >> the first set theory book that came to hand. It's Kunen's Set Theory,
>
> >>
>
> >> an Introduction to Independence Proof. On page 31 he defines cardinal
>
> >>
>
> >> exponentiation just as Michael F. Stemper does! You know, don't you,
>
> >>
>
> >> that the finite cardinals are the same as the natural numbers?
>
> >>
>
> >
>
> > '^' is being used in quite a different way here that may not be perfectly analogous to that usually defined on N.
>
> >
>
>
>
> WRONG!

Then it should be easy to prove 0^0=1 using only natural number arithmetic. How about it?

Dan

Date Subject Author
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Virgil
9/18/13 Dan Christensen
9/18/13 Rotwang
9/18/13 Rock Brentwood
9/18/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/19/13 Virgil
9/19/13 Virgil
9/19/13 Rotwang
9/18/13 Virgil
9/18/13 fom
9/18/13 Rotwang
9/28/13 Shmuel (Seymour J.) Metz
9/29/13 Marshall
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Michael F. Stemper
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/19/13 fom
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Helmut Richter
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 fom
9/19/13 JT
9/19/13 JT
9/19/13 Michael F. Stemper
9/19/13 JT
9/19/13 JT
9/19/13 JT
9/19/13 Helmut Richter
9/28/13 Shmuel (Seymour J.) Metz
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Karl-Olav Nyberg
9/19/13 fom
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/25/13 Rotwang
9/26/13 Dan Christensen
9/27/13 Brian Q. Hutchings
9/19/13 fom
9/18/13 Rock Brentwood
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 Peter Percival
9/20/13 Peter Percival
9/20/13 Dan Christensen
9/20/13 Virgil
9/20/13 Peter Percival
9/20/13 fom
9/20/13 Michael F. Stemper
9/20/13 LudovicoVan
9/21/13 Michael F. Stemper
9/21/13 LudovicoVan
9/21/13 Richard Tobin
9/20/13 Peter Percival
9/20/13 Peter Percival
9/21/13 Dan Christensen
9/19/13 Karl-Olav Nyberg